Funny one this. I put LEDs everywhere in my Firepower with absolutely no problems, so what I'm, suggesting is speculation at best, but I reckon  it might just be right.

What's going on is that the lamp matrix diodes have deteriorated over time, a quick google didn't really reveal anything on diode life expectancy, but the pin is at least 30 years old and for the entire time it's on the diode is doing it's job either in game play or attract mode.  The job of the diode is to block the current flow in one direction. As the diode heads towards failure this ability to block will lessen.

LEDs operate significantly faster than incandescent globes. as they don't have to warm up, and there luminosity is still pretty good at low currents. So a bit of current leakage won't be noticed on an incandescent globe but will (might?) show up on a LED.

SO, if my theory is correct, what I'd do is go to the brightest looking LED that is permanently on and replace the matrix diode associated with that LED. If I'm right the LED should function normally but the others will remain on.  For a few minutes work and a new diode maybe worth a try?  Let us know if this works out.

BTW this does not really provide a satisfactory explanation of why it's only a couple of rows - I have no idea on that. Head box to play field lamps are handled identically I think (it would be possible to change the polling to bias an entire row or column, but I very much doubt this is done).